Geometry of metrics (Q648532)

This paper is a survey of hyperbolic type metrics, which have become popular tools in the study of quasiconformal mappings in the Euclidean \(n\)-space. In his 1988 monograph [Conformal geometry and quasiregular mappings. Lect. Notes Math. 1319. Berlin etc.: Springer (1988; Zbl 0646.30025)] the author based his study of the distortion theory of quasiconformal maps on conformal invariants and proved inequalities for them in terms of hyperbolic type metrics such as the distance ratio metric and the quasihyperbolic metric. During the past decades some new metrics have made their entry to the scene of modern mapping theory such as the Apollonian metric and the Seittenranta metric, and their relation to other metrics has been studied by many people. The author guides the reader through this vast area of research in this easily readable survey. He also points out numerous open problems of varying level of difficulty which may help the interested reader to start research in this topic. Finally, the author discusses some of the solutions to the problems of his earlier survey [``Metrics and quasiregular mappings'', in: Proceedings of the international workshop on quasiconformal mappings and their applications, 2006. New Delhi: Narosa Publishing House. 291--325 (2007; Zbl 1166.30013)].